MATH 423 Linear Algebra II Lecture 28: Inner product spaces.

Transcription

1 MATH 423 Liner Algebr II Lecture 28: Inner product spces.

2 Norm The notion of norm generlizes the notion of length of vector in R 3. Definition. Let V be vector spce over F, where F = R or C. A function α : V R is clled norm on V if it hs the following properties: (i) α(x) 0, α(x) = 0 only for x = 0 (positivity) (ii) α(rx) = r α(x) for ll r F (homogeneity) (iii) α(x + y) α(x) + α(y) (tringle inequlity) Nottion. The norm of vector x V is usully denoted x. Different norms on V re distinguished by subscripts, e.g., x 1 nd x 2.

3 Exmples. V = R n, x = (x 1, x 2,...,x n ) R n. x = mx( x 1, x 2,..., x n ). x p = ( x 1 p + x 2 p + + x n p) 1/p, p 1. Exmples. V = C[, b], f : [, b] R. f = mx f (x). x b ( b 1/p f p = f (x) dx) p, p 1.

4 Inner product: rel vector spce The notion of inner product generlizes the notion of dot product of vectors in R 3. Definition. Let V be rel vector spce. A function β : V V R, usully denoted β(x, y) = x, y, is clled n inner product on V if it is positive, symmetric, nd biliner. Tht is, if (i) x,x 0, x,x = 0 only for x = 0 (positivity) (ii) x, y = y, x (symmetry) (iii) r x, y = r x, y (homogeneity) (iv) x + y,z = x,z + y,z (distributive lw) An inner product spce is vector spce endowed with n inner product.

5 Exmples. V = R n. x,y = x y = x 1 y 1 + x 2 y x n y n. x,y = d 1 x 1 y 1 + d 2 x 2 y d n x n y n, where d 1, d 2,...,d n > 0. x,y = (Dx) (Dy), where D is n invertible n n mtrix. Exmple. V = M m,n (R), spce of m n mtrices. A, B = trce (AB t ). If A = ( ij ) nd B = (b ij ), then A, B = m i=1 j=1 n ij b ij.

6 Exmples. V = C[, b]. f, g = f, g = b b f (x)g(x) dx. f (x)g(x)w(x) dx, where w is bounded, piecewise continuous, nd w > 0 everywhere on [, b]. w is clled the weight function.

7 Inner product: complex vector spce Definition. Let V be complex vector spce. A function β : V V C, usully denoted β(x, y) = x, y, is clled n inner product on V if it is positive, conjugte symmetric, nd depends linerly on the first rgument. Tht is, if (i) x,x 0, x,x = 0 only for x = 0 (positivity) (ii) x, y = y, x (iii) r x, y = r x, y (iv) x + y,z = x,z + y,z (conjugte symmetry) Dependence on the second rgument: x, ry + sz = r x,y + s x,z. (homogeneity) (distributive lw)

8 Exmple. V = C n. x,y = x 1 y 1 + x 2 y x n y n. If z = r + is, then z = r is, zz = r 2 + s 2 = z 2. Therefore x,x = x x x n 2 0. Also, x,y = x 1 y x n y n = x 1 y x n y n = x 1 y x n y n = y,x. Exmples. V = C([, b], C). f, g = f, g = b b f (x)g(x) dx. f (x)g(x)w(x) dx, where the weight function w is bounded, piecewise continuous, nd w > 0 everywhere on [, b].

9 Theorem Suppose x,y is n inner product on vector spce V. Then x,y 2 x,x y,y for ll x,y V. Proof: For ny t C let v t = x + ty. Then v t,v t = x + ty,x + ty = x,x + ty + t y,x + ty = x,x + t x,y + t y,x + tt y,y. Assume tht y 0 nd let t = x,y y,y. Then v t,v t = x,x + t y,x = x,x x,y 2 y,y. Since v t,v t 0 the desired inequlity follows. In the cse y = 0, we hve x,y = y,y = 0.

10 Cuchy-Schwrz Inequlity: x,y x,x y,y. Corollry 1 x y x y for ll x,y R n. Equivlently, for ll x i, y i R, (x 1 y x n y n ) 2 (x x2 n)(y y2 n). Corollry 2 For ny f, g C[, b], ( b 2 b f (x)g(x) dx) f (x) 2 dx b g(x) 2 dx.

11 Norms induced by inner products Theorem Suppose x,y is n inner product on vector spce V. Then x = x,x is norm. Proof: Positivity is obvious. Homogeneity: rx = rx, rx = rr x,x = r x,x. Tringle inequlity (follows from Cuchy-Schwrz s): x + y 2 = x + y,x + y = x,x + y + y,x + y = x,x + x,y + y,x + y,y = x,x + 2 Re x,y + y,y x,x + 2 x,y + y,y x x y + y 2 = ( x + y ) 2.

12 Exmples. The length of vector in R n, x = x1 2 + x x2 n, is the norm induced by the dot product x y = x 1 y 1 + x 2 y x n y n. ( b 1/2 The norm f 2 = f (x) dx) 2 on the vector spce C[, b] is induced by the inner product f, g = b f (x)g(x) dx.

13 Angle Since x,y x y, we cn define the ngle between nonzero vectors in ny rel vector spce with n inner product (nd induced norm): (x,y) = rccos x,y x y. Then x,y = x y cos (x,y). In prticulr, vectors x nd y re orthogonl (denoted x y) if x,y = 0. In complex inner product spce the ngle between vectors is not defined. However the notion of orthogonlity still mkes sense.

14 x x + y y Pythgoren Lw: x y = x + y 2 = x 2 + y 2 Proof: x + y 2 = x + y,x + y = x,x + x,y + y,x + y,y = x,x + y,y = x 2 + y 2.

15 y x y x x + y x y Prllelogrm Identity: x + y 2 + x y 2 = 2 x y 2 Proof: x+y 2 = x+y,x+y = x,x + x,y + y,x + y,y. Similrly, x y 2 = x,x x,y y,x + y,y. Then x+y 2 + x y 2 = 2 x,x + 2 y,y = 2 x y 2.

16 Theorem 1 A norm on vector spce is induced by n inner product if nd only if the Prllelogrm Identity holds for this norm. Theorem 2 (Polriztion Identity) Suppose V is n inner product spce with n inner product, nd the induced norm. (i) If V is rel vector spce, then for ny x,y V, x,y = 1 4( x + y 2 x y 2). (ii) If V is complex vector spce, then for ny x,y V, x,y = 1 4( x + y 2 x y 2 + i x + iy 2 i x iy 2).